Prove that there exist two functions

$$f, g: \mathbb{R} \rightarrow \mathbb{R}$$

such that $f \circ g$ is strictly decreasing, while $g \circ f$ is strictly increasing. (Poland) Andrzej KomisArsKi & Marcin Kuczma

Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties: (1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$; (2) the degree of $f$ is less than $n$. (Hungary) GÉza Kós

A triangle $A B C$ is inscribed in a circle $\omega$. A variable line $\ell$ chosen parallel to $B C$ meets segments $A B$, $A C$ at points $D, E$ respectively, and meets $\omega$ at points $K, L$ (where $D$ lies between $K$ and $E$ ). Circle $\gamma_{1}$ is tangent to the segments $K D$ and $B D$ and also tangent to $\omega$, while circle $\gamma_{2}$ is tangent to the segments $L E$ and $C E$ and also tangent to $\omega$. Determine the locus, as $\ell$ varies, of the meeting point of the common inner tangents to $\gamma_{1}$ and $\gamma_{2}$. (Russia) VASily Mokin & Fedor IvLev

Given a positive integer $n=\prod_{i=1}^{s} p_{i}^{\alpha_{i}}$, we write $\Omega(n)$ for the total number $\sum_{i=1}^{s} \alpha_{i}$ of prime factors of $n$, counted with multiplicity. Let $\lambda(n)=(-1)^{\Omega(n)}$ (so, for example, $\left.\lambda(12)=\lambda\left(2^{2} \cdot 3^{1}\right)=(-1)^{2+1}=-1\right)$.

Prove the following two claims: i) There are infinitely many positive integers $n$ such that $\lambda(n)=\lambda(n+1)=+1 ;$ ii) There are infinitely many positive integers $n$ such that $\lambda(n)=\lambda(n+1)=-1$. (ROMANIA) DAN SchwARZ

For every $n \geq 3$, determine all the configurations of $n$ distinct points $X_{1}, X_{2}, \ldots, X_{n}$ in the plane, with the property that for any pair of distinct points $X_{i}, X_{j}$ there exists a permutation $\sigma$ of the integers $\{1, \ldots, n\}$, such that $\mathrm{d}\left(X_{i}, X_{k}\right)=\mathrm{d}\left(X_{j}, X_{\sigma(k)}\right)$ for all $1 \leq k \leq n$. (We write $\mathrm{d}(X, Y)$ to denote the distance between points $X$ and $Y$.) (United Kingdom) LuKe BetTs

The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2, \ldots, 2011^{2}$, in such a way that every label is used exactly once. We then identify the lefthand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).

Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least M.[4] (ROMANIA) DAN SchwarZ Preamble. For a planar $N \times N$ array, it is folklore that this value is $M=N$, with some easy models shown below. As such, the problem is mentioned in [Béla Bollobás - The Art of Mathematics], 21. Neighbours in a Matrix.

This is not necessarily a flaw on the actual problem, which is presented in a brand novel setting; on the contrary, some general previous knowledge on such type of problems (which we think must be encouraged) is beneficial in searching for the right ideas of a proof.

The idea for a proof goes along the lines of finding a moment in the consecutive filling with numbers of the array, when there are at least $N$ pairs of adjacent filled/yet-unfilled cells (with either distinct filled cells or distinct yet-unfilled cells). Then, when the cell next to that bearing the least label is filled, the difference between its label and the one being filled will be at least $N$.

1	2	$\ldots$	N
$\mathrm{~N}+1$	$\mathrm{~N}+2$	$\ldots$	2 N
$\vdots$	$\vdots$	$\ddots$	$\vdots$
$(\mathrm{~N}-1) \mathrm{N}+1$	$(\mathrm{~N}-1) \mathrm{N}+2$	$\ldots$	$\mathrm{~N}^{2}$

A planar parallel $N \times N$ model array.

1	2	4	$\ldots$		$\mathrm{~N}(\mathrm{~N}-1) / 2+1$
3	5	$\ldots$	$\mathrm{~N}(\mathrm{~N}-1) / 2+2$
6		$\ldots$
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$	$\vdots$
	$\mathrm{~N}(\mathrm{~N}+1) / 2-1$	$\ldots$		$\mathrm{~N}^{2}-2$
$\mathrm{~N}(\mathrm{~N}+1) / 2$		$\ldots$	$\mathrm{~N}^{2}-1$	$\mathrm{~N}^{2}$

A planar diagonal $N \times N$ model array.